Note that if \(x\)  is generic for \(\mu \) , then the definition of weak\(^*\)  convergence and (REF ) imply that
\(A_k\varphi (y)\rightarrow \int \varphi \, d \mu \)  as \(k\rightarrow \infty \) , so every generic point is \(\alpha (\varphi )\) -Birkhoff regular for
every \(\varphi \in \mathcal {C}_b(X)\)  with \(\alpha (\varphi )=\int \varphi \, d \mu \) . In particular, each generic point is quasi-regular.
On the other hand, the space of Borel measures on \(X\)  is weakly sequentially complete by [1]}.
Here, weak sequential completeness means that every fundamental sequence converges (cf. [1]}). A sequence \((\mu _n)_{n\ge 1}\)  of Borel probability measures is fundamental if for every bounded continuous function \(f\) , the sequence \(\int f \text{d}\mu _n\)  is Cauchy. Therefore if \(x\)  is quasi-regular, then the sequence \((\operatorname{\mathcal {E}}(x,n))_{n\ge 1}\)  is fundamental and so converges to a Borel probability measure \(\mu \)  on \(X\) , which is easily seen to be \(T\) -invariant.
